Let G = (V, E) be a graph or digraph and r : V → Z+. An r-detachment of G is a graph H obtained by ‘splitting’ each vertex v ∈ V into r(v) vertices. The vertices v1, ..., vr(v) obtained by splitting v are called the pieces of v in H. Every edge uv ∈ E corresponds to an edge of H connecting some piece of u to some piece of v. Crispin Nash-Williams [9] gave necessary and sufficient conditions for...

For an undirected graph G, an orientation of G is defined as a digraph obtained by replacing each edge with an arc. In this paper, we discuss conditions for G to have an orientation that satisfies given demands on the edge-connectivity between disjoint vertex sets.

Abstract Generalizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity . In particular, together result due to Nash‐Williams Tutte, this implies 7‐edge‐connected bipartite whose edge set decomposes into two edge‐disjoint trees. We best possible as it does not hold for infinitely many 6‐edge‐connected gra...

We survey some recent results on 2-edge and 2-vertex connectivity problems in directed graphs. Despite being complete analogs of the corresponding notions on undirected graphs, in digraphs 2vertex and 2-edge connectivity have a much richer and more complicated structure. It is thus not surprising that 2-connectivity problems on directed graphs appear to be more difficult than on undirected grap...

‎in this paper we defined the vertex removable cycle in respect of the following‎, ‎if $f$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$g in f $‎, ‎the cycle $c$ in $g$ is called vertex removable if $g-v(c)in in f $.‎ ‎the vertex removable cycles of eulerian graphs are studied‎. ‎we also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎

A generalized p-cycle is a digraph whose set of vertices is partitioned in p parts that are cyclically ordered in such a way that the vertices in one part are adjacent only to vertices in the next part. In this work, we mainly show the two following types of conditions in order to find generalized p-cycles with maximum connectivity: 1. For a new given parameter `, related to the number of short...

A flippable edge in an acyclic digraph is an edge whose reorientation leaves the graph acyclic. We characterize the spanning trees T of an undirected graph G such that there exists an acyclic orientation of G whose set of flippable edges is T . In particular for every edge e ∈ E(G) we give a linear algorithm returning an acyclic orientation and a spanning tree T containing e such that T is the ...

In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph Γ, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of Γ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular...